Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Using their own squares, students follow a demonstration on how to fold a five-inch-by-five-inch paper square into halves. They open their paper squares and make four airplane folds, folding each corner into the center. Students then work in pairs to make as many different shapes as they can by folding only along the creases and recording each shape on a sheet of paper. Students develop spatial sense as they investigate the results of subdividing and changing their squares to create different shapes. As they work, students are asked to name the shapes and are given feedback to clarify any misconceptions about the shapes and their sides. The language of geometry, such as square, triangle, five-agon, six-agon, hexagon, and trapezoid, grows naturally from students' explorations and experiences. To conclude the lesson, students share their results with the class. The students' work is taped to a board, and students are asked to identify the geometric shapes and count the number of sides for each shape.
Topics for Discussion
The following areas provide a focus for discussion after you view the video. You may want to customize these areas or focus on your own discussion ideas.
Using Visual Problem Solving
- The task in the video engaged students in paper folding. Describe how this task involved spatial sense and visual reasoning.
- Several children had trouble doing the airplane folds. What does this difficulty tell you about their spatial sense? What other tasks might you plan for students to develop their spatial sense further?
- How can such a task help young children become mathematical problem solvers?
- How appropriate was this task for all students in the classroom? What strategies did Mr. Ramirez use to ensure students' understanding of the task?
- How did Mr. Ramirez promote students' reflection on their own ideas?
Connecting Informal and Formal Mathematical Language
- How were conceptual understanding and language developed from students' experiences in this lesson?
- What were some of the names students created for their shapes (e.g., five-agon, six-agon, a Z with two heads, a car shape)? How did Mr. Ramirez help students connect their informal language to the formal language of mathematics?
- What did you think of the names five-agon and six-agon that were used by some students? How do mathematicians refer to shapes that do not have a special name, in contrast to shapes that do, like a hexagon and a triangle?
- Mr. Ramirez asked if a student's four-sided shape was more a diamond or more a square. The student responded, "Square." How are a diamond and a square similar? How are they different? What role does the concept of orientation play in helping students define a shape?
- One boy in the video was having difficulty counting the number of sides of the shape. How did Mr. Ramirez handle this situation? What other strategies or examples could be used to address this issue?
- In this lesson Mr. Ramirez never told students they were wrong. Identify ways he helped students build on their existing knowledge and integrate new experiences to broaden their understanding.
- Shapes do not necessarily have sides of equal length, i.e., a triangle is not necessarily equilateral. How might you explain this concept for students?
Origami and Other Paper-Folding Activities
Many books and resources on origami and paper folding exist. Investigate the use of origami and paper folding to help students develop spatial sense and formal mathematical language for geometric concepts.